Topics in Combinatorial Group Theory

Abstract

This book presents the main topics of combinatorial group theory (i.e., the theory of groups arising in the form of generators and relations), very accessibly and in the author's transparent, reader-friendly, unpretentious style. The subject is surveyed all the way from its elements up to some of the most recent techniques and results. Either full, very readable, proofs are provided, or else results simply cited without proof though with the necessary background for understanding, if not proving, them given in detail.\par Chapter I surveys the history of the subject, describing the main lines of enquiry that emerged (decision problems, Higman's embedding theorem, varieties of groups, small-cancellation theory, and hyperbolic and automatic groups). In Chapter II the ``Grigorchuk-Gupta-Sidki'' solution of the weak Burnside problem is presented in full, with an application to associative algebras. In Chapter III the basic machinery of combinatorial group theory is introduced, and then in Chapter IV applied to recursive presentability, word problems, and to obtain various other results (notably, that if $G$ is a finitely presented indicable group, then either $G$ is an ascending HNN extension with finitely generated base, or $G$ is virtually a non-trivial generalized free product of two finitely generated groups where the amalgamated subgroup is of finite index in one factor and of arbitrarily large finite index in the other). Chapter V treats of ``geometric representation theory'', in which the geometry of the set $R(G)$ of all representations of a given finitely generated group $G$ in $SL(2,\bbfC)$, considered as an affine algebraic set, is exploited to obtain information about $G$. Here the background algebraic geometry is given in full, after which it is indicated how some familiar results of combinatorial group theory may be proven relatively quickly in this context, as well as remarkable results such as that of the author and Shalen according to which any finitely presented group of deficiency $\geq 2$ can be decomposed as an amalgamated product of two groups with the amalgamated subgroup of index $\geq 2$ in one factor and $\geq 3$ in the other. In Chapter VI generalized free products and HNN extensions are examined in detail and various results based on these constructions are proved (e.g. the Adyan-Rabin theorem on Markov properties, and the algorithmic undecidability of the problem of determining whether or not any finitely presented group $G$ has $H\sb 2(G;\bbfZ) = 0$). Finally, in Chapter VII, a complete exposition of the structure theorem for a group acting without inversions on a tree is given, following {\it J. P. Serre}'s book ``Trees'' (Zbl 0548.20018) in the main. From this the subgroup theorems of Schreier-Nielsen, Kurosh, and Hanna Neumann-Karrass- Solitar are deduced quickly. The chapter, and the book, ends by showing that $SL(2,F)$, $F$ a field with a discrete valuation, acts without inversions on a tree, whence Ihara's theorem that $SL(2,F)$ is a (specific) amalgamated product is inferred.\par The wide range of topics considered and the extraordinary clarity of exposition make this work an excellent introduction to combinatorial group theory, accessible to graduate students in algebra, but also a very good read for the expert.